
Brazilian Journal of Physics, vol. 34, no. 1, March, 2004 279

Hadron-Hadron Interactions in Coulomb Gauge QCD

Sérgio Szpigel,
Faculdade de Cîencias Bioĺogicas, Exatas e Experimentais, Universidade Presbiteriana Mackenzie

Rua da Consolac̃ao 930, 01302-907 S̃ao Paulo, Brasil

G. Krein, and R. S. Marques de Carvalho
Instituto de F́ısica Téorica, UNESP, Rua Pamplona 145, 01405 São Paulo, Brasil

Received on 15 August, 2003.

We describe the derivation of an effective Hamiltonian which involves explicit hadron degrees of freedom and
consistently combines chiral symmetry and color confinement. We use a method known as Fock-Tani (FT)
representation and a quark model formulated in the context of Coulomb gauge QCD. Using this Hamiltonian,
we evaluate the dissociation cross section ofJ/ψ in collision withρ.

1 Introduction

The experimental observation ofJ/ψ suppression in ultra-
relativistic heavy-ion collisions by NA38 [1] and more re-
cently the anomalousJ/ψ suppression in Pb+Pb collisions
observed by NA50 [2] have attracted much attention as a
possible signal for a quark-gluon plasma (QGP) [3].

Such a suppression can be described by phenomenolog-
ical models either in a QGP [4] or in a hadronic scenario [5].
Several theoretical studies have been described [6] and the
subject is still controversial. In this way, microscopic ap-
proaches that allows one to consistently treat hadron-hadron
interactions in terms of the underlying quark-gluon structure
would provide a useful tool for the understanding of this is-
sue.

In a previous work [7] we described a field theoretical
method known as Fock-Tani (FT) representation used to de-
rive an effective Hamiltonian involving explicit hadron de-
grees of freedom and its application to study hadron inter-
actions using a nonrelativistic microscopic quark model. In
this paper we consider the extension of the method to a mi-
croscopic relativistic quark model formulated in the context
of Coulomb gauge QCD which consistently combines chi-
ral symmetry and color confinement[8]-[10]. Our aim is to
set up an effective calculational scheme to comprehensively
investigate hadronic structure and interactions such as char-
monium suppression.

2 Coulomb Gauge QCD

The canonical QCD Hamiltonian in the Coulomb gauge,
∇ ·A = 0, can be written as [11]-[13]:

H =
∫

dx ψ† (−iα · ∇+ mqβ)ψ − g

∫
dx ψ†α ·Aψ

+
1
2

∫
dx

(J−1Π · JΠ + B ·B)

+
1
2

∫
dx dyJ−1ρa(x)Kab(x,y;A)J ρb(y), (1)

wheremq is the current quark mass,J = det(∇ ·D) is the
Faddeev-Popov determinant andDab = δab∇ + igfabcAc

is the covariant derivative in the adjoint representation.
The termKab is the non-Abelian Coulomb kernel

Kab(x,y;A) ≡ 〈x, a| g

∇ ·D (−∇2)
g

∇ ·D |y, b〉, (2)

whereρa is the full color charge density given by

ρa(x) = ρa
g(x) + ρa

q (x)

= fabcAb(x) ·Πc(x) + ψ†(x)
λa

2
ψ(x). (3)

Note that in the Abelian limit,D → ∇, the QED Coulomb
interaction is recovered.

K → −g2〈x, a|1/∇2|y, b〉 = g2δab/4π|x− y|. (4)

The dynamical degrees of freedom are the transverse gauge
fieldsAa, the transverse conjugate gluon momentaΠa and
the quark fieldψ.

The key features of the Coulomb gauge are [14]: a)
The elimination of non-dynamical degrees of freedom cre-
ates a long-range instantaneous non-Abelian Coulomb in-
teraction, which provides a confinement scenario: infrared
divergences make colored states infinitely heavy, removing
them from the physical spectrum; color neutral states, on
the other hand, remain physical; b) The absence of spurious
degrees of freedom yields Fock states with positive normal-
izations. This is essential to build nonperturbative models
for the QCD vacuum and a quasiparticle basis of constituent
quarks and gluons.

3 Quark Model with Chiral Symme-
try Breaking

The starting point of our model is an approximate QCD
Hamiltonian in the Coulomb gauge, in which we use an ef-


280 Sérgio Szpigel

fective Coulomb kernelKab(x,y;A) → V (x − y) as ob-
tained in Ref. [14]. This is obtained makingJ → 1 and
neglecting quarks in Eq. (3),ρa

q (x) = 0. The derivation
is based on a self-consistent method to construct a gluonic
quasiparticle basis. The kernel can be interpreted as an ef-
fective interaction between two heavy quarks and the results
remarkably well with lattice computations [15]. For long
distances, the numerical results forV (x − y) are almost
identical toV (x− y) = σ|x− y| and is, therefore, infrared
singular and needs in general a careful regularization when
dealing with numerical simulations. In the present paper, for
simplicity of explaining the model and methods employed
to construct an effective hadron-hadron interaction, we use
a simpler form forV (x−y) (see below). However, it should
be clear that the methods developed here are not dependent
on the specific choice of the kernel.

With such a kernel, the general form of the model Hamil-
tonian in the fermionic sector is:

Ĥ =
∫

dx[Ĥ0(x) + ĤI(x)], (5)

whereĤ0 is the Hamiltonian density of the Dirac field op-
eratorψ(x),

Ĥ0(x) = ψ†(x) (mqβ − iα.∇) ψ(x), (6)

andĤI is an effective instantaneous interaction term

ĤI(x) =
1
2

∫
dyψ†(x)

λa

2
ψ(x) V (x−y) ψ†(y)

λa

2
ψ(y).

(7)
The next step consists in constructing an approximate

new vacuum state for the Hamiltonian in the form of a pair-
ing ansatz [8, 10]. Let’s first define a “trivial” vacuum|0〉
throughb0

fsc|0〉 = d 0
fsc|0〉 = 0, whereb0 andd0 are quark

annihilation operators, in terms of which the quark field op-
erator is given by

ψ(x) =
∫

dp
(2π)3/2

[
u0

s(p)b0
s(p) + v0

s(p)d0†
s (−p)

]
ei p.x,

(8)
where color and flavor indices have been neglected. Then, a
nontrivial vacuum|0̃〉 can be defined through a Bogoliubov-
Valatin transformation such asb|0̃〉 = d|0̃〉 = 0, where theb
andd quark annihilation operators are related to the bare op-
eratorsb0 andd0 by the BVT. In terms of the dressed quark
operators, the quark field operator can be expanded as

ψ(x) =
∫

dp
(2π)3/2

[
us(p)bs(p) + vs(p)d†s(−p)

]
ei p.x,

(9)
with the quasiparticle spinorsus, vs given in terms of theu0

s
andv0

s spinors as [8, 10]

us(p) =
1√
2

[√
1 + sin ϕ(p) +

√
1− sin ϕ(p)p̂.~α

]
u0

s,

vs(p) =
1√
2

[√
1 + sin ϕ(p)−

√
1− sin ϕ(p)p̂.~α

]
v0

s ,

(10)

whereϕ(p) is sometimes called the chiral angle and is de-
termined by a gap equation (see below).

The normal order of the Hamiltonian relatively to the
new vacuum gives:

Ĥ = H0 + Ĥ2 + ĤA
2 + Ĥ4, (11)

whereH0 is a constant and gives the energy of the new vac-
uum, and

Ĥ2 =
∫

dpE(p)
[
b†s(p) bs(p) + d†s(−p) ds(−p)

]
, (12)

whereE(p) is the energy of a free quark:

E(p) = sinϕ(p) A(p) + cosϕ(p) B(p),

A(p) = mq +
2
3

∫
dk V (p− k) sin ϕ(p),

B(p) = p +
2
3

∫
dk V (p− k) cos ϕ(p) p̂.k̂, (13)

and

Ĥ4 =
1
2

∫
dp dk dq V (q)

(
λa

c1c2
λa

c1c2

4

)

×
4∑

j,l=1

: Θj
c1c2

(p,p + q) Θl
c3c4

(k,k− q) :,(14)

gives 10 different terms that are combinations of the follow-
ing four vertices (here we have introduced the color indices
for clarity):

Θ1
c′c(p,p′) ≡ u†s′(p

′)us(p) b†s′c′(p
′)bsc(p),

Θ2
c′c(p,p′) ≡ −v†s′(p

′)vs(p) d†sc(−p)ds′c′(−p′),

Θ3
c′c(p,p′) ≡ u†s′(p

′)vs(p) d†s′c′(p
′)d†sc(−p),

Θ4
c′c(p,p′) ≡ v†s′(p

′)us(p) d†s′c′(−p′)bsc(p). (15)

The termĤA
2 is the anomalous, nondiagonal Bogoliubov

term. In order to bring the single-quark Hamiltonian into a
diagonal form, one has to requirêHA

2 = 0, which leads to
the gap equation

A(p) cos ϕ(p)−B(p) sin ϕ(p) = 0. (16)

It is useful to introduce a running quasiparticle quark
mass,M(p), through the equations

cos ϕ(p) =
p

E(p)
, sin ϕ(p) =

M(p)
E(p)

(17)

with E(p) =
√

p2 + M2(p). One can identify an effective
constituent quark mass asMq = max[M(p)] and extract it
from the low momentum behavior of the chiral angle [16].

4 Effective Hadron-Hadron Hamil-
tonian

Effective hadron-hadron potentials in quark potential mod-
els have been obtained within several early approaches such


Brazilian Journal of Physics, vol. 34, no. 1, March, 2004 281

as adiabatic methods [17], resonating group [18], variational
techniques [19] and the QBD formalism [20]. In this work
we use the Fock-Tani (FT) formalism, which was devel-
oped independently by Girardeau [21] and Vorob’ev and
Khomkin [22] in the context of atomic physics and has re-
cently been extended to hadronic physics [7]. The method
shares some similarities with Weinberg’s quasi-particle ap-
proach [23].

In the following, we present the main features of the
Fock-Tani formalism for the derivation of an effective
meson-meson interaction. We start by specifying the mi-
croscopic Hamiltonian in Fock space (F):

H = T (µ) q†µqµ + T (ν) q†νqν +
1
2
Vqq (µν;σρ) q†µq†νqρqσ

+
1
2
Vqq (µν; σρ) q†µq†νqρqσ + Vqq (µν; σρ) q†µq†νqρqσ.(18)

In Eq.(18),T is the kinetic energy andVqq, Vqq and Vaq

are respectively the quark-quark, antiquark-antiquark and
quark-antiquark interactions. The indicesµ, ν, · · · repre-
sent spatial, color, spin, and flavor quantum numbers of the
quarks and antiquarks and a summation over repeated in-
dices is implied. The quark and antiquark operators obey
standard anticommutation relations:

{qµ, q†ν} = {qµ, q†ν} = δµν ,

{qµ, qν} = {qµ, qν} = {qµ, q†ν} = 0. (19)

A generic meson state inF , composed by a quark-
antiquark pair, is denoted by|α〉, whereα represents the
meson quantum numbers (c.m. momentum, internal energy,
spin and flavor). Such a state can be written as:

|α〉 = M†
α|0〉 ≡ Φµν

α q†µq†ν |0〉, (20)

whereM†
α is the meson creation operator,Φµν

α is the me-
son wave function and|0〉 is the vacuum state, defined as
qµ|0〉 = qν |0〉 = 0. Using the quark anticommutation rela-
tions of Eq. (19), and the orthonormalization condition for
the Φ’s, one can show that the meson operators satisfy the
following noncanonicalcommutation relations:

[Mα,M †
β ] = δαβ −∆αβ , [Mα,Mβ ] = 0, (21)

where∆αβ = Φ∗µν
α Φµσ

β q†σqν + Φ∗µν
α Φρν

β q†ρqµ is the term
that manifests the composite nature of the mesons.

The change to the FT representation is implemented by
means of a unitary transformationU , such that asinglecom-
posite meson state|α〉 is transformed into asingle ideal-
meson state|α) = m†

α|0) ≡ U−1|α〉, wherem†
α andmα

are the ideal-meson creation and annihilation operators that
satisfy canonical commutation relations:

[mα,m†
β ] = δαβ , [mα,mβ ] = [m†

α,m†
β ] = 0. (22)

By definition, them† andm commute with the quark
and antiquark operators. In this way, within the FT represen-
tation one recovers the possibility of using traditional field
theoretic techniques such as Wick’s theorem, Feynman dia-
grams, etc.

The operatorU is constructed as a power series in the
bound state wavefunctionsΦ. Once the operatorU is
known, one proceeds by transforming the original quark-
model operators, such as currents and Hamiltonian. This
is accomplished by transforming initially the quark and an-
tiquark operators and substituting these into the expressions
of quark model operators. The explicit form ofU and the
derivation of the transformed quark and antiquark operators
is discussed in detail in Ref. [7].

The structure of the transformed Hamiltonian is:

HFT = Hq + Hm + Hmq . (23)

The quark Hamiltonian,Hq, has an identical structure to
the one of the microscopic quark Hamiltonian of Eq. (18),
except that the term corresponding to the quark-antiquark
interaction is modified such that it does not produce the
quark-antiquark bound states.Hmq describes quark-meson
processes as meson breakup into a quark-antiquark pair, etc.
The term involving only ideal meson operators,Hm, has a
component that represents an effective meson-meson inter-
action:

Hmm = Φ∗µν
α H(µν; µ′ν′)Φµ′ν′

β m†
αmα

+
1
2

∑

αβγδ

Vmm(αβ; γδ)m†
αm†

βmδmγ , (24)

where the effective meson-meson potentialVmm is a sum of
several different terms involvingH(µν; µ′ν′) and the prod-
uct of four wave-functions corresponding to the initial and
final meson states.

Note that the effective meson Hamiltonian is model in-
dependent, in the sense that it depends only on the general
forms of the microscopic quark Hamiltonian and of the me-
son states.

5 Ongoing Calculations

In order to illustrate the application of the framework
through a simple example, we have calculated the scattering
cross section for charmonium dissociation by inelastic scat-
tering onρ mesons, using the effective meson-meson Hamil-
tonian derived in section 4 and the quark model Hamiltonian
with chiral symmetry breaking described in section 3. Our
final aim is to perform the calculation using the potential
derived from the gauge sector of the Coulomb gauge QCD
Hamiltonian, as in Ref. [14]. However, such an interaction
exhibits a strong singularity atq → 0 that needs to be reg-
ulated in the process of performing a numerical integration.
We are still in the process of regulating such a numerical sin-
gularity (there is no real singularity since the integrands are
finite atq = 0). Thus, here we just show the results obtained
using a Gaussian interaction given by:

V (q) =
1

(2π)3/2
V0 (8πχ)

3
2 e−2χq2

. (25)

The J/ψ mesons are composites of a heavy quark and
a heavy antiquark pair, denoted by(QQ), and theρ mesons
are composites of a light quark and a light antiquark, de-
noted by(qq). The final mesonsD, D are composites of a
(qQ) or a (Qq) pair and can be either in the fundamental


282 Sérgio Szpigel

D, D(11S0) or in the excitedD∗, D
∗
(31S1) states. The ex-

plicit form of the creation operator for a composite meson
is

M†
CSF (p) =

∑

csf

Cc1c2
C χs1s2

S Ff1f2
F

∫
dk1dk2Φp(k1,k2)

× q†c1s1f1
(k1)q

†
c2s2f2

(k2), (26)

whereCC , χS , andFF are respectively the color, spin and
flavor Clebsch-Gordan coefficients. For the spatial meson
wave-function we employ a Gaussian ansatz :

Φk1k2
p = δ(3)(p− k1 − k2)

(
b2

π

) 3
4

e−b2k
2
/2, (27)

wherek = ηk1 − (1− η)k2, with η = m2/(m1 + m2) and
b is the Gaussian parameter related to the r.m.s. radius of the
meson by< r2 >=

√
3/2 b.

There are six final state reaction channels for the reac-
tion, allowed by momentum conservation:

J/ψ(31S1) + ρ(31S1) → D(1S) + D(1S). (28)

The total cross section for the reaction is a function of
the center-of-mass energy and is obtained by summing over
all possible final channelsσtot(s) =

∑6
f=1 σfi(s). For nu-

merical evaluations, the parameter values used are:

mQ = 1.67 GeV, mq = 0.33 GeV,

V0 = 0.5 GeV, χ = 1.0 GeV −2,

bQQ = 0.560 GeV, bqq = 0.380 GeV,

bQq = bqQ = 0.440 GeV.

0 0.25 0.5 0.75 1
Ec.m. (GeV)

0

2

4

6

8

σ 
(m

b)

Total cross section
J/Ψ + ρ −> D + Dbar (Stot=0)
J/Ψ + ρ −> D + D

*
bar or D

*
 + Dbar (Stot=1)

J/Ψ + ρ −> D
*
 + D

*
bar (Stot=2)

J/Ψ + ρ −> D
*
 + D

*
bar (Stot=1)

J/Ψ + ρ −> D
*
 + D

*
bar (Stot=0)

Figure 1. Cross-sections forJ/ψ + ρ scattering.

In Fig. 1 we show the cross sections for the reaction as
a function of the relative kinetic energy of theJ/ψ and the
ρ in the center-of-mass system.

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